Word on the street, or at least in the 'ibles in the Related panel ----->

is that you should replace the inferior nickel-cadmium (NiCad) cell, with a new and improved nickel-metal-hydride (NiMH). The NiMH cells have pretty much the same voltage, but greater capacity, no memory effect, and are made from materials that are less toxic.

That's a good summary of the whole NiMH vs NiCad saga.

BTW, there's no guarantee that the factory cell is an inferior NiCad. It could be an inferior NiMH.

Also it may be the case that your cheap solar cells are not providing enough energy, per day, to give you the number of hours of light at night that you desire.

Anyway if you just want to replace that battery, then seek out some new, name brand (e.g. Duracell(r), Energizer(r), etc.) AA, NiMH cells. They're easy enough to find in Earth's finer hardware and department stores. People use 'em in their flashlights, and cameras, and whatnot, so they're found in that part of the store.